Power-Law Scaling of the Impact Crater Size-Frequency Distribution on Pluto: a Preliminary Analysis Based on First Images from New Horizons’ Flyby

Home , Neukum (Martian crater), Tombaugh (crater)

Volume12(2016) PROGRESSINPHYSICS Issue1(January)

Power-Law Scaling of the Impact Crater Size-Frequency Distribution on Pluto: A Preliminary Analysis Based on First Images from New Horizons’ Flyby

Felix Scholkmann Research Office of Complex Physical and Biological Systems (ROCoS), Bellariarain 10, 8038 Zürich, Switzerland E-mail: [email protected] The recent (14th July 2015) flyby of NASA’s New Horizons spacecraft of the dwarf planet Pluto resulted in the first high-resolution images of the geological surface- features of Pluto. Since previous studies showed that the impact crater size-frequency distribution (SFD) of different celestial objects of our solar system follows power-laws, the aim of the present analysis was to determine, for the first time, the power-law scaling behavior for Pluto’s crater SFD based on the first images available in mid-September 2015. The analysis was based on a high-resolution image covering parts of Pluto’s regions Sputnik Planum, Al-Idrisi Montes and Voyager Terra. 83 impact craters could be identified in these regions and their diameter (D) was determined. The analysis revealed that the crater diameter SFD shows a statistically significant power-law scaling (α = 2.4926 ± 0.3309) in the interval of D values ranging from 3.75 ± 1.14 km to the largest determined D value in this data set of 37.77 km. The value obtained for the scaling coefficient α is similar to the coefficient determined for the power-law scaling of the crater SFDs from the other celestial objects in our solar system. Further analysis of Pluto’s crater SFD is warranted as soon as new images are received from the spacecraft.

1 Introduction [14–16] and Mercury [17]. The first close-up images of Pluto from NASA’s New Hori- Due to the lack of high-resolution images available, it has zons spacecraft, received in mid-September 2015, show a not been possible until now to analyze the impact crater SFD complex surface structure of Pluto never seen before in this of Pluto. With the first images now available from NASA’s detail. During the spacecraft’s flyby of Pluto on 14th July New Horizons mission, the aim of the present paper was to 2015, images were taken by New Horizons’ Long Range Re- conduct such a first, preliminary, analysis. connaissance Imager (LORRI) with a cooled 1024 × 1024 2 Materials and methods pixel CCD camera from a distance of approx. 12,500 km making it possible to obtain high-resolution images of Pluto’s 2.1 Data surface. Due to the slow transmission (about 1–2 Kbps), it For the present analysis the raw images∗ obtained by the New will take around 16 months for all flyby images of Pluto to be Horizons’ LORRI as of 14th September 2015 were visually received in full [1]. inspected in order to find an image showing impact craters of Many phenomena in astrophysics follow a power-law, i.e. Pluto with the highest resolution possible. An additional se- the relationship between features exhibits a scale-invariance. lection criterion was that the image had to be taken by LORRI Examples are the characteristics of the channel network on at an angle capturing the region of interest maximally paral- Mars [2], the relationship between solar flare occurrence and lel to the camera, minimizing geometrical distortions of the total flare energy [3], the correlation between a supermassive features in the image. black hole mass and the host-galaxy bulge velocity dispersion The image lor 0299174809 0x630 sci 4 (in the following (“M-sigma relation”) [4], the distribution of initial masses for denoted as LOR-0299174809)was selected as fulfilling these a population of stars (“initial mass function”) [5], Kepler’s criteria (see Figure 1(b)). LOR-0299174809 displays a par- third law, or the distribution of galaxies in the universe [6–8]. ticular area covering parts of Pluto’s regions Sputnik Planum, Size-frequency distributions (SFD) of natural objects also Al-Idrisi Montes and Voyager Terra. The image was taken by follow in general a power-law. Examples are the SFD of frag- LORRI on 14th July 2015, 10:14:50 UTC, with an exposure ment sizes due to a fragmentation process [9], the SFD of time of 150 ms. landslides [10], the particle SFD of volcanic ash [11], the mass distribution objects of the Kuiper belt [12] — or the 2.2 Determination of crater diameter values SFD of impact crater diameters on celestial objects. Already in 1940 Young showed that the impact crater SFD The image LOR-0299174809was analyzed in Adobe Illustra- on the Earth’s Moon can be described by two power-laws tor (version CS5; Adobe Systems, CA, USA) by first visually with different scaling exponents. Further studies extended ∗LORRI Images from the Pluto Encounter, http://pluto.jhuapl. the analysis to other celestial objects, e.g. Earth [13], Mars edu/soc/Pluto-Encounter

26 F. Scholkmann. Power-Law Scaling of the Impact Crater Size-Frequency Distribution on Pluto Issue1(January) PROGRESSINPHYSICS Volume12(2016)

identifying the craters on the image and measuring their di- than for the empirical values, a p-value is calculated. For ameters (D). The obtained values were than rescaled to give p < 0.1 the fit of the power-law model to the empirical data the final values in the unitkm. To do so, the informationgiven is considered to be statistically not significant, i.e. it can be on NASA’s website∗ was used. Information on the website in- ruled out that the empirical distribution obeys a power-law dicated that image number three (from top), which covers the scaling. Thus, the p-value in this case represents a measure region displayed in LOR-0299174809, is 470 km in width. of the hypothesis that is tested for validity. A high value of p corresponds to a good fit. 2.3 Statistical analysis 3 Results, discussion and conclusion For the statistical analysis we used the mathematical framework provided by the Santa Fe Institute [18,19]. The data 83 impact craters could be identified and their diameter val- were analyzed in Matlab (version 2010a; Mathworks Inc., ues were determined, ranging from 0.84 km to 37.77 km. Natick, MA, USA). Using the obtained 83 D values and the methods described in Section 2.3.1, the scaling coefficient α was determined to be α = −2.4926 ± 0.3309 and the scaling threshold value 2.3.1 Estimation of the lower-bound and exponent of the D = . ± . D power-law model to be min 3 75 1 14 km. Thus, for in the range [3.75 ± 1.14 km, 37.77 km] the values follow a power-law. A quantity x shows power-law scaling if it stems from a prob- The log-likelihood (L) of the data D > Dmin under the fitted ability distribution p(x) ∼ x−α, with the exponent α defining power law was determined to be L = 104.5688. the characteristics of the scaling. To test if an empirically The statistical test, as described in Section 2.3.2, employ- obtained probability distribution follows a power-law, classi- ing 10,000 synthetic data sets revealed a p value of 0.2241. cally a histogram is calculated and the distribution is analyzed Thus, according the test the hypothesis cannot be refuted that on a doubly logarithmic plot. Since p(x) ∼ α ln(x) + const., the data follows a power-law, i.e. Pluto’s crater SFD shows a a power-law distributed quantity x follows a straight line in power-law scaling. Figure 1(c) visualizes the power-law be- the plot. Besides the fact that this method was (and is still) havior of the crater SFD. used to investigate power-law scaling of different quantities How do the results of the presented analysis relate to the this approach can generate significant systematic errors [18]. findings about characteristics of the crater SFD of other celes- Therefore, for the present analysis we used a framework pre- tial objects? As mentioned in the introduction, it well estab- sented by Clauset et al. [18] that circumvents these errors and lished that the crater SFD of all investigated celestial bodies also offers the possibility of estimating the lower bound of in our solar system exhibit a power-law scaling. the power-law (xmin). The determination of xmin is crucial For example, according to an analysis performed by when analyzing empirical data for power-law scaling since Robertson and Grieve from 1975 the crater SFD of the Earth often the power-law behavior applies only for the tail region is characterized by α ≈ −2 (for D > 8 km) [13]. An own of the distribution, i.e. for values greater than the threshold analysis using the updated data of impact craters on Earth † value xmin. (n = 188) based on the Earth Impact Database revealed For the obtained crater diameter (D)data(= x) the power- α = 2.0286 (for D > 7 km). The EarthMoon’scrater SFD has law threshold value Dmin (= xmin) was determined based on been intensively investigated since the 1940’s when Young the method described by Clauset et al. [18] which uses the [20] initially showed that for large D the crater SFD follows Kolmogorow-Smirnov (KS) statistics. The scaling exponent a power-law with α = −3, and for small D the scaling is de- α was then calculated with a maximum likelihood fitting scribed by α = −1.5. Subsequent studies described the scal- method also described by Clauset et al. [18]. ing with laws governed by α = 2 (for D = [∼ 2 km, 70 km]) [21], as well as α = −2 (for D < 100 m) and α = −2.93 > 2.3.2 Determination of the statistical significance of the (for D a few 100 m) [22], for example. Further stud- power-law fit ies showed that the scaling-relations of the lunar crater SFD need to include the observation that multiple power-laws are In order to determine if the fitted power-law can be consid- necessary to describe the whole SFD spectrum, i.e. α de- ered statistically significant, a goodness-of-fit test described pends on D [23, 24]. A solution for optimally fitting the by Clauset et al. [18] was employed. To this end, power-law crater SFD was described based on the idea of using a poly- distributed synthetic data was generated with values of α and nomial function to fit the SFD data in the log-log space, i.e. th xmin that are equal to the values obtained by fitting the em- it could be shown that a polynomial function of 7 degree pirical data to the power-law model. Each synthetic data set fit the data well for D = [300 m, 20 km]. The polynomial is than fitted to the model and the KS statistics determined. function included an extra term accounting for the fact that Based on the occurrences of times the KS statistic is larger the scaling function also depends on the geological charac-

∗http://tinyurl.com/n9k5mmc †http://www.passc.net/EarthImpactDatabase

F. Scholkmann. Power-Law Scaling of the Impact Crater Size-Frequency Distribution on Pluto 27 Volume12(2016) PROGRESSINPHYSICS Issue1(January)

teristics of the region investigated — a finding also made by other studies (e.g., [24–26]). For an extended range of D, in later work a 11th degree polynomial function was published by Neukum [27] valid for D = [10 m, 300 km] and covering scaling exponents in the range of α = [−1, 4]. For the Martian satellites Phobos and Deimos, the crater SFD was determined as being described by a power-law with α ≈ 1.9 for D = [44 m, 10 km] [16]. Thus, the finding of the present analysis concerning the power-law characteristics (i.e., α = 2.4926 ± 0.3309 for D = [3.75 ± 1.14 km, 37.77 km]) of the crater SFD of Pluto is comparable to the power-laws observed for the other celestial bodies. That Pluto’s diameter scaling for D < 3.75 ± 1.14km does not follow the α = −2.4926 scaling relies most prob- ably on the fact that small craters are much faster deterio- rated due to erosion and that counting of craters with small D was not perfectly possible due to the limited resolution of the available image. The lowest D value (3.75 ± 1.14 km) for which the power law holds might interpreted as related to a transition from simple to complex craters. Interestingly, such a “transition diameter” was predicted for Pluto to be in the range of 4–5 km [28–30]. This analysis, of course, should be regarded only as a preliminary study for further follow-up as soon as the full set of images from Pluto is available and the images have been pro- cessed to deliver a high-resolution picture of Pluto’s surface morphology. A limitation of the present analysis is that only one high-resolution image with sufficient craters was available. It was therefore only possible to obtain a relatively low number of crater diameter values (n = 83). Knowledge of Pluto’s crater SPF will not only give in- sights in the universality of the crater SFD scaling relations but necessarily will also help in the understanding of the history of Pluto and the characteristics of the Kuiper belt which Pluto is part of.

Acknowledgements I thank Rachel Scholkmann for proofreading the manuscript, and the New Horizonsproject team at the Johns HopkinsUni- versity Applied Physics Laboratory for discussion concerning the copyright status of the publicly released LORRI images.

Submitted on October 15, 2015 / Accepted on November 6, 2015

References Fig. 1: (a) View of Pluto taken in July 2015 by LORRI on board 1. Rienzi G. How exactly does New Horizons send all that data back NASA’s New Horizons spacecraft. In the field of view the west- from Pluto? Available from: http://hub.jhu.edu/2015/07/17/ new-horizons-data-transmission ern lobe of the Tombaugh region is depicted. (b) LORI image lor0299174809 0x630 sci 4 showing a particular area covering 2. Caldarelli G., De Los Rios P., Montuori M., Servedio V.D.P. Statistical features of drainage basins in mars channel networks. The European parts of Pluto’s regions Sputnik Planum, Al-Idrisi Montes and Voy- Physical Journal B, 2004, v. 38 (2), 387–391. ager Terra. (c) Visualization of the power-law scaling of the impact 3. Hudson H.S. Solar flares, microflares, nanoflares, and coronal heating. crater size distribution. P(D): complementary cumulative distribu- Solar Physics, 1991, v. 133 (2), 357–369. tion function; D: crater diameter. Images (a) and (b) were obtained 4. Ferrarese L., Merritt D. A fundamental relation between supermassive from NASA, Johns Hopkins University Applied Phsics Laboratory, black holes and their host galaxies. The Astrophysical Journal, 2000, Southwest Research Institute. v. 539 (1), L9–L12.

28 F. Scholkmann. Power-Law Scaling of the Impact Crater Size-Frequency Distribution on Pluto Issue1(January) PROGRESSINPHYSICS Volume12(2016)

5. Kroupa P., Weidner C., Pflamm-Altenburg J., Thies I., Dabringhausen 27. Neukum, G. Meteoritenbombardement und Datierung planetarer J., Marks M., Maschberger T. The stellar and sub-stellar initial mass Oberflächen. Habilitationsschrift. 1983, München, Germany: Univer- function of simple and composite populations. In Planets, Stars and sität München. Stellar Systems, Oswalt T.D., Gilmore G. (Editors), 2013, Springer 28. Moore J.M., Howad A.D., Schenk P.M., McKinnon W.B., Pappalardo Netherlands, 115–242. R., Ryan E., Bierhaus E.B., et al. Geology before Pluto: Pre-encounter 6. Watson D.F., Berlind A.A., Zentner A.R. A cosmic coincidence: The considerations. Ikarus, 2014, v. 246, 65–81. power-law galaxy correlation function. The Astrophysical Journal, 29. Bray V.S., Schenck P.M. Pristine impact crater morphology on Pluto — 2011, v. 738 (1), 1–17. Expectations for New Horizons. Icarus, 2015, v. 246, 156–164. 7. Baryshev Y.V., Labini F.S., Montuori L., Pietronero L., Teerikorpi P. 30. Zahnle K., Schenk P., Levison H., Donnes L. Cratering rates in the outer On the fractal structure of galaxy distribution and its implications for Solar System. Icarus, 2003, v. 163, 263–289. cosmology. Fractals, 1998 v. 6 (3), 231–243. 8. Joyce M., Sylos Labini F., Gabrielli A., Montuori M., Pietronero L. Ba- sic properties of galaxy clustering in the light of recent results from the Sloan Digital Sky Survey. Astronomy & Astrophysics, 2005, v. 443 (1), 11–16. 9. Turcotte D.L. Fractals and fragmentation. Journal of Geophysical Re- search, 1986, v. 91 (B2), 1921–1926. 10. Stark C.P., Hovius N. The characterization of landslide size distributions. Geophysical Research Letters, 2001, v. 28 (6), 1091–1094. 11. Wohletz K.H., Sheridan F., Brown W.K. Particle size distributions and the sequential fragmentation/transport theory applied to volcanic ash. Journal of Geophysical Research, 1989, v. 94 (B11), 15703–15721. 12. Bernstein G.M., Trilling D. E., Allen R. L., Brown K. E., Holman M., Malhotra R. The size distribution of transneptunian bodies. The Astro- physical Journal, 2004, v. 128 (3), 1364–1390. 13. Robertson P.B., Grieve R.A.F. Impact structures in Canada — Their recognition and characteristics. The Journal of the Royal Astronomical Society of Canada, 1975, v. 69, 1–21. 14. Bruckman W., Ruiez A., Ramos E. Earth and Mars crater size frequency distribution and impact rates: Theoretical and observational analysis. arXiv:1212.3273, 2013. 15. Barlow N.G. Crater size-frequency distributions and a revised martian relative chronology. Icarus, 1988, 75, 285–305. 16. Thomas, P. Veverka, J. Crater densities on the satellites of Mars. Icarus, 1980, v. 41, 365–380. 17. Strom R.G., Chapman C.R., Merline W.J., Solomon S.C., Head J.W. Mercury cratering record viewed from Messergner’s first flyby. Science, 2008, v. 321 (5885), 79–81. 18. Clauset A., Shalizi C.R., Newman M.E.J. Power-law distributions in empirical data. SIAM Review, 2009, v. 51 (4), 661–703. 19. Virkar Y., Clauset A. Power-law distributions in binned empirical data. Annals of Applied Statistics, 2014, v. 8 (1), 89–119. 20. Young, J., A Statistical Investigation of Diameter and Distribution of Lunar Craters. Journal of the British Astronomical Association, 1940, v. 50, 309–326. 21. Cross C.A. The size distribution of lunar craters. Monthly Notices of the Royal Astronomical Society, 1966, v. 134, 245–252. 22. Shoemaker E.M., Hait M.H., Swann G.A., Schleicher D.L., Dahlem D.H., Schaber G.G., Sutton R.L. Lunar regolith at tranquillity base. Science, 1970, v. 167 (3918), 452–455. 23. Chapman C.R., Haefner, R.R. A critique of methods for analysis of the diameter-frequency relation for craters with special application to the Moon. Journal of Geophysical Research, 1967, v. 72 (2), 549–557. 24. Neukum G., König B., Arkani-Hamed J. A study of lunar impact crater size-distributions. The Moon, 1975, v. 12 (2), 201–229. 25. Baldwin R.B. On the history of lunar impact cratering: The absolute time scale and the origin of planetesimals. Icarus, 1971, v. 14, 36–52. 26. Head J.W., Fassett C.I., Kadish S.J., Smith D.E., Zuber M.T., Neu- mann G.A., Mazarico E. Global distribution of large lunar craters: implications for resurfacing and impactor populations. Science, 2010, v. 329 (5998), 1504–1507.

F. Scholkmann. Power-Law Scaling of the Impact Crater Size-Frequency Distribution on Pluto 29